sum over all possible wave-vectors and two possible orthogonal polarization states . The operators and are the photon creation and annihilation operators for each indexed mode of the field. The simplicity of the Jaynes–Cummings model comes from suppressing this general sum by considering only a single mode of the field, allowing us to write where the subscript indicates that we are considering only the resonant mode of the cavity. is the dipole atom-field interaction Hamiltonian (here is the position of the atom). Electric field operator of a quantized electromagnetic field is given by and dipole operator is given by . Setting and making the definition where the s are the orthonormal field modes, we may write where and are the raising and lowering operators acting in the subspace of the atom. The application of the Jaynes-Cummings model allows suppression of this sum, and restrict the attention to a single mode of the field. Thus the atom-field Hamiltonian becomes: . Rotating frame and rotating-wave approximation Next, the analysis may be simplified by performing a passive transformation into the so-called "co-rotating" frame. To do this, we use the interaction picture. Take . Then the interaction Hamiltonian becomes: We now assume that the resonance frequency of the cavity is near the transition frequency of the atom, that is, we assume . Under this condition, the exponential terms oscillating at are nearly resonant, while the other exponential terms oscillating at are nearly anti-resonant. In the time that it takes for the resonant terms to complete one full oscillation, the anti-resonant terms will complete many full cycles. Since over each full cycle of anti-resonant oscillation, the net effect of the quickly oscillating anti-resonant terms tends to average to 0 for the timescales over which we wish to analyze resonant behavior. We may thus neglect the anti-resonant terms altogether, since their value is negligible compared to that of the nearly resonant terms. This approximation is known as the rotating wave approximation, and it accords with the intuition that energy must be conserved. Then the interaction Hamiltonian (taking to be real for simplicity) is: With this approximation in hand (and absorbing the negative sign into ), we may transform back to the Schrödinger picture: Jaynes-Cummings Hamiltonian 2 Using the results gathered in the last two sections, we may now write down the full Jaynes-Cummings Hamiltonian: The constant term represents the zero-point energy of the field. It will